*3.1. Primary Forecasting Step*

As the primary forecasting step, the created data matrix with 13 variables was fed into the NSGAII algorithm. The employed fitness function is an MLPNN with one hidden layer whose hidden layer size and transfer functions are 7, tansig, and tansig, respectively. Different combinations of the 13 variables (input vectors) were generated, and their fitness values were evaluated by the NSGAII in each iteration. Non-dominated solutions are the outputs of this step. Each output is a MLPNN, which contains the best non-dominated input vector as its input. The Pareto front of the NSGAII related to the generated non-dominant solutions is shown in Figure 7.

**Figure 7.** Pareto front of the NSGAII.

To demonstrate the importance of the feature selection, forecasting results of the best input vector generated by NSGAII was compared to the 13 variables 1 variable (*x* − 1) input vectors, using MLPNN as the forecasting model. The results are presented in Table 1. Explanation of the employed error indicators can be found in Appendix A.


**Table 1.** Forecasting results for different input sets using MLPNN.

As seen in Table 1, using selected input vector generated by the NSGA II results in better forecasting accuracy. Furthermore, to compare the forecasting accuracy of the MLPNN and ANFIS models, the selected input vector was used as the input of the ANFIS models with different training algorithms. The results are given in Table 2.


**Table 2.** Forecasting results for each model.

Comparing Table 1 to Table 2 it can be observed that using selected input vector, MLPNN model has a better forecasting accuracy compared to the ANFIS model. In addition, among tested ANFIS training algorithms, GA demonstrates better performance. Response surface of output versus input 1 and input 2 related to hybrid learning algorithm and GA learning algorithm (meta-heuristic with the best performance) are given in Figure 8 where input 1 and input 2 are (*x* − 1) and (*x* − 2) respectively.

**Figure 8.** Response surface of output versus input 1 and input to for hybrid (**left**) and GA (**right**) learning algorithms.
